Two states hydrogenlike model for high-order harmonic generation and an isolated attosecond pulse generation in a He⁺ ion

Abstract

We present an analytic quantum theory of high harmonic generation by low frequency laser fields for a two-state hydrogenlike atom model. In the case of an initial coherent superposition of two states, the theory clearly explains why high conversion efficiency can be obtained. Furthermore, we apply this model to generate an intense isolated attosecond pulse by the proposed combined field. By mixing an infrared chirped pulse with a fundamental chirped pulse, a supercontinuum spectrum can be formed. The long quantum trajectory is suppressed, and the short one is enhanced and an intense isolated 95-as pulse is obtained by He⁺ ion. Our simulation shows that the produced pulses with good signal-to-noise ratio are obtained without any phase compensation. In particular, these results are analyzed by using both classical and time-frequency analyses.

Introduction

High-order harmonic generation (HHG) is a rapidly developing topic in the field of laser–atom interaction. Besides its fundamental interest, it represents an attractive technique for the production of ultrafast coherent radiation in the UV and soft-x-ray regions of the spectrum using a tabletop system. The HHG procedure can be well understood by the semiclassical three-step model [1] and the quantum model [2]. According to the three-step model, first the electron tunnels through the barrier formed by the Coulomb potential and the laser field, then it oscillates quasi freely driven by the laser field and acquires additional kinetic energy, and finally it can recombine with the parent ion and emit radiation. The harmonic spectrum is characterized by a rapid drop at low orders followed by a broad plateau where all the harmonics have the same strength and a sharp cut-off around harmonic energy I_p+3.17U_p, where I_p is the atomic ionization potential and U_p is the ponderomotive energy, i.e., the cycle-averaged kinetic energy of an electron gained in a mono-chromatic laser field. It has been theoretically and experimentally demonstrated that an attosecond pulse train with a periodicity of half an optical cycle can be generated from HHG by a multicycle driving laser pulse [3], [4], and every pulse in the train contains two peaks originating from the short and long quantum paths and characterized by an electron travel time in the continuum of approximately half and one optical cycles, respectively. For practical applications, an isolated attosecond pulse is preferable to a chain of attosecond pulses. The main problem so far has been to generate sufficiently intense HHG pulses for practical applications. The most efficient methods depend on a single-atom signal, which is then phase matched in the target medium [5], [6]. Most models of HHG in single-atom level are based on the Lewenstein model [2] which is often referred to as the strong-field approximation of HHG. In the Lewenstein model, the single atomic state was taken into account. We extend this model to two states' cases, analytically. This method does allow us to study some of the features that are unique for HHG from atoms or ions within a well-established framework. Examples of the importance of including the second state and underlying physics of the HHG, are preparing the initial state as a superposition of the ground and the first excited states or resonant interaction between bound atomic states etc. We can simply apply the extracted result of our model and obtain the corresponding dipole momentum functions. The driven two-level system is a particularly important model in physics, since it has proven to be very useful in describing many aspects of the interaction of matter with an electromagnetic field. By considering some excited state during the laser pulse, the conditions for achieving high conversion efficiency can be obtained.

The superposition state can be obtained by using one harmonic pulse with the frequency corresponding to the energy difference between the two bound states [7] before the fundamental laser pulse.

Before proceeding, we note that harmonic generation by preparing the initial state as a coherent superposition of two bound states was first proposed by Gauthey et al. [8]. Watson and co-workers [9] demonstrated that a harmonic spectrum with distinct plateaus could be obtained by such superposition states. Wang and co-workers made a numerical study of high-order harmonic generation in a He⁺-like model ion when the initial state is prepared as a coherent superposition of the ground state and an excited state [10]. Miloševic [11] formulated a quantum theory of high-order harmonic generation by a strong laser field in the presence of more bound states by means of Green's operator. In this approach, the bound-state depletion is not obtained by solving the Schrödinger equation, as a result, the bound state amplitudes are not coupled together and also it is not explicitly seen from the presented derivation the dynamic of the wave packet that contributes to the high harmonic-generation process.

In this paper, we present an appropriate approach to extending the traditional harmonic generation (Lewenestein model in Ref. [2]) to two states hydrogenlike atom model. Our model not only gives the explicit expression for the time-dependent wave function with the extracted probability amplitudes for the bound and continuum states, but also provides clear explanations of the physical mechanisms of HHG for two states hydrogenlike atom model.

On the other hand, due to the inherent characteristic of the plateau, the modulation of the harmonics is high, which limits the application of generating single attosecond pulse. Hence, a lot of effort has been devoted to generate intense isolated attosecond pulses. Recently, Chen and co-workers numerically generated sub-40-as pulses from a coherent superposition state from a He⁺ ion in a two-color laser field [12]. In this article we continue the same effort for generating short and intense isolated attosecond pulse. For practical application of our theory we study the control of high-order harmonics cutoff and as-pulse generation by combining a fundamental chirped laser with a controlling field laser by using a coherent superposition of the ground state and the first excited state as the initial state. With the introduction of the controlling field laser to the fundamental one, the trajectories of the electron wave packet are modified, the short quantum trajectory is enhanced, the long quantum trajectory is suppressed, and the cutoff position of the harmonics can be remarkably extended. By superposing a properly selected range of harmonic spectrum in the continuum region, an intense isolated 95-as pulse can be generated. To explore the underlying mechanism responsible for the enhancement of the harmonic efficiency and the cutoff extension, we perform time–frequency analysis of emitted pulses by means of the wavelet transform of the induced dipole acceleration and a classical simulation based on the three-step model.

This paper is organized as follows. Section 2 contains a general presentation of extended theory and a discussion of its quasiclassical interpretation, supplemented by argument of the underlying physics. We present general expressions for harmonic strengths. In particular, for example, we study the case of 1s and 2s states. In next section we generate the intense isolated attosecond pulse by means of our model. Section 4 sums up our conclusions. Atomic units ($\hslash =e=m_e=a_0=1$) are used throughout, unless stated. Since we aim to describe a rather general way of possible control over the harmonic emission, we perform our calculations for a simple hydrogenic atom, He⁺ ion. At this point it is worth pointing out that the results we present are not directly related to the structure of the atomic potential and therefore can be straightforwardly extended to any type of ion or atom.